So i was given a question stated in the title and I have to show this for
A)$2x+7y=3$
B)$3x+ 8y = 3$
C)$4x + 9y = 5$
I understand how to use the linear diophantine equation to solve these questions but what confuses me is the "using the mod 4 arithmetic" in the question. What exactly does that mean and how would you do it?
If you have a system of equations it implies more than one equation and that you are supposed to find a solution that satisfies all of them simultaneously. I presume the system consists of those three equations, not that you are supposed to "show this for" each of them individually.
What they mean is to reduce both sides modulo $4$ and see what it gives.
$2x + 3y \equiv 3 \pmod{4}$.
$3x \equiv 3 \pmod{4}$.
$y \equiv 1 \pmod{4}$.
You can easily check that $x \equiv 1 \pmod{4}$ based on the second equation but that and the third equation contradict the first.